If the Boolean expression (p ⇒ q) ⇔ (q ∗ (∼ p)) is a tautology, then the Boolean expression p ∗ (∼ q

Q: Wait, why does a tautology mean the two sides are equivalent?

That is a fantastic conceptual question! A tautology is a statement that is always true. If we have an expression $A \Leftrightarrow B$ that is a tautology, it means that for every possible truth value of $p$ and $q$, the statement is true. This can only happen if $A$ and $B$ always have the same truth value. Therefore, they are logically equivalent. Think of it as two different paths leading to the exact same destination.

Q: How do I know the $*$ operator isn't something exotic like NAND or XOR?

Logic is elegant, not chaotic. In these problems, we look for the simplest mapping that satisfies the equivalence. By substituting the implication identity, we see a direct structural match. If it were XOR or NAND, the truth tables would not align with the OR operator structure we derived. Trust the algebra—it rarely lies!

Q: Why do we flip $p \lor \sim q$ to $\sim q \lor p$ at the end?

Because the options are given in the form of implications ($A \Rightarrow B$). The standard form for an implication is $\sim A \lor B$. By rearranging, we force the expression into a format that matches the answer choices. It is like rearranging a room to fit the furniture perfectly.

Question

If the Boolean expression $(p \Rightarrow q) \Leftrightarrow (q * (\sim p))$ is a tautology, then the Boolean expression $p * (\sim q)$ is equivalent to

Options:

(A) $q \Rightarrow p$

(B) $\sim q \Rightarrow p$

(C) $p \Rightarrow \sim q$

(D) $p \Rightarrow q$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given that the statement $(p \Rightarrow q) \Leftrightarrow (q * (\sim p))$ is a tautology. A tautology is a statement that is universally true, meaning the logical expressions on both sides of the double implication $\Leftrightarrow$ must be equivalent.

Therefore, we can establish the fundamental equivalence:
$$(p \Rightarrow q) \equiv (q * (\sim p))$$

### The Rosetta Stone of Logic

To simplify the left-hand side, we utilize the standard identity for logical implication. An implication $p \Rightarrow q$ is logically equivalent to the disjunction of the negation of the antecedent and the consequent.

This is expressed as:
$$(p \Rightarrow q) \equiv (\sim p \lor q)$$

By substituting this identity into our master equation, we obtain:
$$(\sim p \lor q) \equiv (q * (\sim p))$$

### The Detective Work

We now compare the two sides of the equation to identify the unknown operator $*$. Recall that the logical OR operator ($\lor$) is commutative, meaning the order of operands does not change the truth value.

We can rewrite the left-hand side as:
$$(q \lor (\sim p)) \equiv (q * (\sim p))$$

By direct comparison, it is evident that the unknown operator $*$ is the **logical OR operator**, denoted by $\lor$.

### Final Calculation

The problem asks us to evaluate the expression $p * (\sim q)$. Substituting our discovered operator $\lor$ into this expression, we get:
$$p \lor (\sim q)$$

To match the standard format of logical implications, we apply the commutative property:
$$(\sim q \lor p)$$

Finally, applying the implication rule in reverse, where $(\sim A \lor B) \equiv (A \Rightarrow B)$, we identify $A$ as $q$ and $B$ as $p$. This yields the final result:
$$q \Rightarrow p$$

**The final logical expression is $q \Rightarrow p$.**

Animated Solution for Mathematics - Sets and Relations: If the Boolean expression $(p \Rightarrow q) \Leftrightarrow (q * (\sim p))$ is a tautology, then the Boolean expression $p * (\sim q)$ is equivalent to

Select Answer:

Visualized Solution

Understanding the Tautology

The Implication Identity

Equating LHS and RHS

Applying Commutative Law

Identifying the Operator $*$

Evaluating the Target Expression

Substituting the Operator

Rearranging for Comparison

Final Equivalence

The Sigma Insight: Types of Sets and Set Operations

Analyzing the Setup

The Rosetta Stone of Logic

The Detective Work

Final Calculation

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